Bring a large pot of water to a boil. Meanwhile, break two eggs into a warm, large and shallow bowl (like a lasagna dish!). Mix them together with ¼ cup of Pecorino Romano and ¼ cup of grated parmesan. Set aside.
Salt the boiling water and throw in the spaghetti. Cook until al dente, 7-9 minutes.
While the pasta is cooking, warm the 4 tablespoons of olive oil in a medium frying pan over medium heat. Add the brussel sprouts and sprinkle in a pinch of salt and pepper. Cook for 6-7 minutes, until the sprouts are starting to crisp and brown. Add the garlic and prosciutto, and continue to cook for 4-5 more minutes. Add the ½ cup of white wine and continue to sauté until the wine reduces by half.
Drain the pasta, leaving a bit of water clinging to it. Add the noodles straight to the bowl with the eggs, and toss the noodles quickly. The egg mixture will quickly cling to the noodles, so be sure to get them evenly coated!
Pour in the brussel sprouts and prosciutto mixture onto the the spaghetti and toss. Top the bowl with the chopped parsley, red pepper flakes, and salt and pepper to taste. Enjoy!